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MATH STUDENT BOOK 10th Grade | Unit 1
804 N. 2nd Ave. E.Rock Rapids, IA 51246-1759
800-622-3070www.aop.com
MATHSTUDENT BOOK
10th Grade | Unit 1
LIFEPAC Test is located in the center of the booklet. Please remove before starting the unit.
MATH 1001A Mathematical System
INTRODUCTION |3
1. UNDEFINED TERMS 5POINTS |5LINES |6PLANES |7SELF TEST 1 |9
2. BASIC DEFINITIONS 11A DEFINITION OF DEFINITIONS |11GEOMETRIC TERMS |12SELF TEST 2 |17
3. GEOMETRIC STATEMENTS 19POSTULATES |19THEOREMS |25SELF TEST 3 |32GLOSSARY |35
Unit 1 | A Mathematical System
|1
804 N. 2nd Ave. E. Rock Rapids, IA 51246-1759
© MCMXCVII by Alpha Omega Publications a division of Glynlyon, Inc. All rights reserved. LIFEPAC is a registered trademark of Alpha Omega Publications, Inc.
All trademarks and/or service marks referenced in this material are the property of their respective owners. Alpha Omega Publications, Inc. makes no claim of ownership to any trademarks and/or service marks other than their own and their affiliates, and makes no claim of affiliation to any companies whose trademarks may be listed in this material, other than their own.
Author: Milton R. Christen, M.A.
Editor-in-Chief: Richard W. Wheeler, M.A.Ed.
Consulting Editor: Robert L. Zenor, M.A., M.S.
Revision Editor: Alan Christopherson, M.S.
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A Mathematical System | Unit 1
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ObjectivesRead these objectives. The objectives tell you what you will be able to do when you have successfully completed this LIFEPAC©. When you have finished this LIFEPAC, you should be able to:
1. Identify, draw, and label models of undefined terms and defined terms.
2. Define some basic geometric terms.
3. Define postulate and theorem, and recognize a model of each.
4. Describe the four items that are involved in a mathematical system.
A mathematical system is a logical study of shape, arrangement, and quantity. Algebra, geometry, trigonometry, and calculus are examples of mathematical systems. Geometry is the logical study of the shape and size of things. The word geometry comes from the Greek and means earth measurement.Any mathematical system contains four items:
� Basic undefined terms.
� All other terms carefully defined.
� Some basic properties that are accepted without proof. (These are called postulates.)
� All other properties that are established by proof. (These are called theorems.)
A Mathematical System
Introduction
Unit 1 | A Mathematical System
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Survey the LIFEPAC. Ask yourself some questions about this study and write your questions here.
A Mathematical System | Unit 1
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1. UNDEFINED TERMSIn geometry there are three terms that we do not attempt to define. They are point, line, and plane. These three terms are the foundation upon which geometry is built.
Section ObjectiveWhen you have completed this section, you should be able to:
1. Identify, draw, and label models of undefined terms and defined terms.
POINTS A point cannot be seen. It has no size, no shape, no color, no physical properties. It is an imagi-nary thing. We do, however, represent a point on paper by a dot. A point has position or location.
To be able to talk about a point, we name it with a capital letter and show it on paper as a dot.
Five locations or positions on this page are points and are named A, B, C, D, and E. More than five points are on the page. In fact, the number of points on this page is infinite.
Complete the following activities.
1.1 We think of a point as a _____________________________________________________________________ .1.2 A point is represented by ___________________________________________________________________ .1.3 Place names on these points.
• • •
••
1.4 How many points are located on this page? ________________________1.5 What physical characteristics do points have? ________________________________________1.6 Have you ever seen a point? _____________________
•D•
• •
•
A
B E
C
Unit 1 | A Mathematical System
Section 1 |5
LINESA line can be thought of as a collection of points that are lined up one after the other and extend infinitely far in opposite directions. A line in geometry is straight, not curved or bent. Since there are an infinite number of points on this page they can be lined up to form an infinite number of lines.
We represent a line on paper like this:
The arrowheads on the ends tell us the line extends indefinitely. Since a line is made up of points, it has no thickness or other physical characteristics. A line is named by naming two of its points and placing a dou-ble-pointed arrow above the two letters.
To the right are four lines named EB, LD, DW, and MB. A line may also be named by using a single lower-case letter as illustrated.
) ) ))
Supply the information required.
1.7 We think of a line as a collection of __________________________________________________ .
1.8 Unless otherwise indicated, a line in geometry is understood to be (straight, curved,either)
_____________________________ .
1.9 Name the following lines.•
•• •
••
E
DB
AC
F
a. __________________________ b. _________________________ c. _________________________
1.10 How many lines are contained on this page? ______________________________________
1.11 What is indicated by arrowheads on a line? _________________________________
• ••
•
••
••
m
E BD
D
WM
B
L
n a
b
A Mathematical System | Unit 1
6| Section 1
PLANESA plane can be thought of as a collection of points in a flat surface that extends in all direc-tions without stopping. To represent a plane, a figure is drawn like the ones illustrated. (Remember that the edges are not really there.) A plane is named by using the name of one of its points. Thus we have plane A, plane B, and plane C. As with points and lines, planes do not have any physical properties. A billion planes could be stacked in a pile and still not have enough thickness to be measured. A plane is defined by at least three points not all on the same line.
Write the following information.
1.12 A plane can be thought of as a (flat, curved) a. ___________________________ surface, made up of
b. ______________________________ .
1.13 Name the following planes.
R•
••
ST
a. __________________________ b. _________________________ c. __________________________
1.14 How long is a plane? __________________________________________________________________________
1.15 How thick is a plane? __________________________________________________________________________
1.16 Can planes be seen? _____________________
C
•
•
•
A
B
Unit 1 | A Mathematical System
Section 1 |7
We may also use models to represent points, lines, and planes. Representing geometric shapes in some tangible way will help to illus-trate some of their mutual relationships.
The sharp end of a pin, tack, or pencil, for example, can be a model for a point. A pencil, meter stick, or soda straw can represent a line. A table top, sheet of cardboard, or wall in a room can be a model for a plane.
Here are some ways points, lines, and planes are related. Note the terms used in talking about points, lines, and planes.
Points A and B are on line l. Line l contains points A, B, and C. C is a point of AB.
Lines a and b intersect at point P. Q is not a point of line b.
Plane M and plane N intersect at AB. Their intersection is AB. Point C is in plane M but not in plane N. Line t intersects plane N at point D. Plane M contains line r. Every point of line r is in plane M.
Points A and B are in both planes M and N.
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TEACHER CHECKinitials date
Review the material in this section in preparation for the Self Test. The Self Test will check your mastery of this particular section. The items missed on this Self Test will indicate specific areas where restudy is needed for mastery.
•
••
•
•
• •
••
•A
a
•P
Q
b
BC
l
C
A
B
N
M
pr
t
D
A Mathematical System | Unit 1
8| Section 1
Name the undefined terms for which each is a model (each answer, 3 points).
1.01 ________________________________
1.02 _________________________________
1.03 _________________________________
1.04 _________________________________
1.05 _________________________________
1.06 _________________________________
Match the description in Column II with its model in Column I (each answer, 2 points).
Column I Column II
1.07 ________________
1.08 ________________
1.09 ________________
1.010 ________________
1.011 ________________
1.012 ________________
SELF TEST 1
•
• T
•S
B
R
M•
N•
m
Figure 1 Figure 2 Figure 3
Figure 4 Figure 5 Figure 6
a. MN
b. Point T
c. Plane R
d. Plane S
e. Point B
f. Line m
)
Unit 1 | A Mathematical System
Section 1 |9
Refer to the model and complete the statements (each answer, 2 points).
•
••
•
•
•
••
B
A
w
x
T
C
R
Q
E
P
S •
1.013 The intersection of plane S and plane R is ________________________ .
1.014 Line PQ a. ___________________ plane S at point b. ___________________ .
1.015 Line w and line x intersect at ___________________ .
1.016 Points B and R are in plane ___________________ .
1.017 Name three lines shown to be in plane S.
a. ________________________ b. ______________________ c. _______________________
1.018 Line w lies in plane ___________________ .
1.019 Name all the points shown to be in plane S. ____________________________
1.020 Name all the points shown to be in plane R. ____________________________
SCORE TEACHERinitials date
4252
A Mathematical System | Unit 1
10| Section 1
804 N. 2nd Ave. E.Rock Rapids, IA 51246-1759
800-622-3070www.aop.com
MATHSTUDENT BOOK
ISBN 978-0-86717-631-5
9 7 8 0 8 6 7 1 7 6 3 1 5
MAT1001 – Apr ‘15 Printing
